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The space-time fractional diffusion equation with Caputo derivatives

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Abstract

We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or Riemann-Liouville) derivative of order β∈(0, 2] and the first-order time derivative with Caputo derivative of order α∈(0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Xiamen University, 361005, Xiamen, China

    F. Liu

  2. School of Mathematical Sciences, Queensland University of Technology, 4001, Qld., Australia

    F. Liu

  3. School of Mathematical Sciences, South China University of Technology, 510640, Guangzhou, China

    F. Huang

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  1. F. Huang
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Correspondence to F. Liu.

Additional information

Fawang Liu received his MSc from Fuzhou University in 1982 and PhD from Trinity College, Dublin, in 1991. Since graduation, he has been working in computational and applied mathematics at Fuzhou University, Trinity College Dublin and University College Dublin, University of Queensland, Queensland University of Technology and Xiamen University. Now he is a Professor at Xiamen University. His research interest is numerical analysis and techniques for solving a wide variety of problems in applicable mathematics, including semiconductor device equations, microwave heating problems, gas-solid reactions, singular perturbation problem, saltwater intrusion into aquifer systems and fractional differential equations.

Fenghui Huang received her MSc and PhD from Xiamen University, Xiamen, China, in 2001 and 2004. Now She is a lecturer at South China University of Technology. Her research interest is numerical computation for PDE, especially, solving the variety of problems in the Computational Fluid Dynamics, such as incompressible fluid flow and turbulence. She also pay respect to some applied problems in fractional differential equations.

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Huang, F., Liu, F. The space-time fractional diffusion equation with Caputo derivatives. JAMC 19, 179–190 (2005). https://doi.org/10.1007/BF02935797

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  • DOI: https://doi.org/10.1007/BF02935797

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